Chapter P, Problem 5RE

In Exercises 5 and 6, simplify the expression. Assume that denominators are not zero.

5. $\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}$

Solution

Simplify the given expression.

  $\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}=v^4$

Given information:

The given interval is:

  $\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}$

Formula used:

To simplify an expression, follow the steps:

1. Simplify the numerator and denominator.
2. Factor out the common terms.
3. ${\left(a^m\right)}^n=a^{m\times n}$
4. ${\left(ab\right)}^n=a^n{b}^n$

Calculation:

In order to simplify the given expression:

  $\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}$

First simplify the numerator by taking its cube:

  $\begin{array}{c}\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}=\frac{u^3{\left(v^2\right)}^3}{v^2{u}^3};\text{ since }{\left(ab\right)}^n=a^n{b}^n\\ =\frac{u^3{v}^{2\times 3}}{v^2{u}^3};\text{ since }{\left(a^m\right)}^n=a^{m\times n}\\ =\frac{u^3{v}^6}{v^2{u}^3}\end{array}$

Now factor out the common terms from numerator and denominator, as:

  $\begin{array}{c}\frac{u^3{v}^6}{v^2{u}^3}=\frac{u^3{v}^2{v}^4}{v^2{u}^3}\\ =\frac{u^3{v}^2\left(v^4\right)}{v^2{u}^3}\\ =\frac{\cancel{u^3{v}^2}\left(v^4\right)}{\cancel{v^2{u}^3}}\\ =\left(v^4\right)\\ =v^4\end{array}$

Thus, the simplified form of given expression is:

  $\frac{{\left(u{v}^2\right)}^3}{v^2{u}^3}=v^4$